Routing and Rate Allocation to Provide End-to-End
Delay Guarantees under PGPS Scheduling
Aniruddha S. Diwan
Kumar N. Sivarajan
ECE Dept, IISc Bangalore
diwan@ece.iisc.ernet.in
Tejas Networks, Bangalore
kumar@tejasnetworks.com
Abwucf- This paper considers the problem of routing sessions with
Quality of Service (QoS) requirements in a network under Packet Generalized Proceqsor Sharing (PGPS) scheduling. PGPS is a non-preemptive
scheduling policy that tracks GPS. The GPS policy operates by allocating
a weight @Ffor a session n whose traffic uses link m. These weights determine the rate at which the traffic from session n is served at link m
and the rate in turn determines the end-to-end delay of packets belonging
to session n. As a deterministic and easily computable end-to-end delay
bound is available for locally stable sessions, we consider the locally stable regime in t h k paper. Two separate problems are considered in this
paper. The first problem deals with the practically important inverse procedure of specifying appropriate weights for sessions at each link on their
paths, that satisfy predetermined delay bounds, when the set of sessions
to be routed is given. Here we show that the fixed routing case can be
formulated as a Iirieur progrum (LP) and the adaptive routing case can he
(MILP). The second problem
formulated as a mixed rrireger lrrieur ~~r(igrurn
examines the performance of PGPS scheduling policy when providing persession QoS guarantees. We measure the performance in terms of weighred
u r r r e d rrufic.. We derive an upper hound on the weighted carried traffic
for mil heuristic algorithm for admission control that operates within the
locally stable domain. This upper bound can be obtained by computing
a linear program (LP). By simulating a simple heuristic algorithm for admission control, we show that this upper bound is reasonably tight. Hence
our upper bound can be used as a metric against which the performance
of different algorithms can he compared.
1. INTRODUCTION
The rise in the popularity of the World Wide Web (WWW)
has resulted in the Web becoming the Internet’s killer application and today the Web occupies more than 75% of Internet backbone traffic. However, because of the time sensitivity
of audiohideo and the huge bandwidth requirements of video,
network delivery of audiohide0 posed considerable technical
challenge. Until fairly recently, the only practical ways to get
musichide0 were to play from a CD-ROM or to download a
very large file across the Internet for playback at the user’s
desktop. But with the emergence of new technologies like audiohideo streaming and with the tremendous increase in the
link speed (e.g., optical fibers), it is now possible to deliver
audiohideo over the Internet directly to desktop. These developments have given rise to a plethora of real-time applications
that provide users a richer and more realistic experience of multimedia. Soon streaming audiohideo will occupy large portion
of the Internet traffic.
However, real-time multimedia requires continuous availability of sufficiently high bandwidth in the channel. This is
especially important for time-critical traffic like streaming audiohideo. Variable audiolvideo packet delays can cause annoying audio starts and stops and picture jitters. The availability of
very high-speed links hasn’t solved this problem because of the
explosive growth of the Internet. Today, an estimated 100 to
This work was carried out at the Indian Institute of Science and supported by
a research grant from Nortel Networks.
0-7803-7097-1/01/$10.00 02001 IEEE
200 million users have access to the Internet. This calls for various network considerations and one of these is the problem of
provisioning of Quality of Service (QoS) to the real-time services. Good QoS provides a guaranteed bandwidth at a constant
small delay even under congested conditions.
The only way to guarantee QoS is by allocating resources
(like bandwidthhuffer) at intermediate routers on per-session
basis and schedulirig the packets at the router interfaces so that
a session gets its share of the router resources. The scheduling policy employed at the router interfaces governs the end-toend delay of the packets. Traditional FIFO scheduling doesn’t
bound this end-to-end delay and it doesn’t provide isolation
as a session can block other sessions by sending a continuous
stream of packets. Various scheduling policies have been proposed in the literature to provide per-session end-to-end delay
(and throughput) guarantees in high-speed networks. Among
these, the class of schemes based on the Generalized Processor
Sharing (GPS) policy are most popular in the literature. GPS
is an idealized fluid discipline with a number of desirable properties such as minimum rate guarantees with perfect isolation
among sessions and deterministic and easily computable endto-end delay bounds. Because these properties are important
in the context of guaranteeing QoS in high-speed networks,
the GPS discipline has become a basis for an entire range of
scheduling policies that are related to GPS.
GPS scheduling was first proposed in [ 13, [2l. In [31 and [SI,
end-to-end delay bounds are derived when the session traffic
is (a,p)-constrained [4]. Work on analysis of GPS for statistically bounded arrivals is reported in [ 5 ] and [6]. Many scheduling policies have been proposed which aim at approximating
GPS fairness in a packetized environment [ 2 ] , [7]. In this paper we consider the simplest scheduling policy called Packet
GPS (PGPS) [ 2 ] .PGPS is a non-preemptive scheduling policy
that tracks GPS. GPS (or PGPS) works by assigning a session
a weight q5 at each link on the session’s route. The weight of a
session at a link on its path decides its share of the link bandwidth at that link which in turn decides the end-to-end delay.
Most of the work on GPS has addressed the problem of obtaining end-to-end delay bound for prespecified session weights.
The first problem that we investigate in this paper deals with
the practically important inverse procedure of determining appropriate weights for sessions at each link on their paths, that
satisfy prespecified delay bounds, when the set of sessions to
be routed is given. All the sessions are assumed to be leaky
bucket constrained. When we assume that a session route is
prespecified, it becomes the case of fixed routing. We show
that the fixed routing case can be formulated as a linear program
(LP). However, prespecifying a route limits the admissibility of
2128
11. GPS SCHEDULING
POLICY
Fig. I . Hypothetical example to illustrate advantage of adaptive routing. All
links are 15 Mbps and all sessions require 5 Mbps.
Consider N sessions contending for a link 1. The GPS
scheduling policy assigns N positive real numbers (weights) ,
4; ,4\, . . . ,& for the N sessions. This is called a GPS assignment for sessions at the server of link 1. The weights signify the
relative amount of service that each session gets, i.e., if $ ( T , t )
is defined as the amount of session i traffic served by the GPS
server at link 1 during an interval [ T , t ] ,then
Sf(Tt )
>&
(1)
si(T,t ) - 4;
a session because enough resources may not be available on
that route. At the same time, another route for the same session may exist with enough resources, which is not considered
in the fixed routing framework. This in turn limits the network
schedulable region which calls for determining a session route
instead of prespecifying one. This can considerably increase
the network schedulable region. We present a motivating example below.
for any session i that is continuously backlogged in the interval
[~,tA
] . session is backlogged at time t if a positive amount
of that session's traffic is queued at time t. Note from (1) that
whenever session i is backlogged it is guaranteed a minimum
service rate of
Fig 1 shows a simple network consisting of 4 nodes and 4
links. Assume each link has a bandwidth of 15 Mbps. Assume
(a,b) and (c, d ) are the only source-destination pairs (SD-pairs)
on which sessions can be present. Assume all sessions require a bandwidth of 5 Mbps. Now consider the fixed routing
case. Suppose the route for ( a ,b) is a -+ c -+ b and that for
( c , d ) is c -+ b -+ d. It can be easily seen that the maximum number of sessions that the network can support simultaneously are ( 3 , 0 ) ,(2, l),(1,2) or ( 0 , 3 ) ,where the first number indicates the maximum on ( a ,b ) and the second number
indicates the maximum on (e,d ) . Now consider the adaptive
routing case. There are two routes for each SD-pair now. The
routes for ( a ,b ) are a -+ c -+ b and a -+ d + b and that
for ( c , d ) are c -+ b -+ d and c + a -+ d. In this case the
the maximum number of sessions supported by the network are
(6,0),(4, a), (2,4) or (0,6). Thus the schedulable region of
fixed outing case is a subset of the schedulable region of adaptive routing.
where rr is the rate of the link. This rate is called the session
i backlog clearing rate since a session i backlog of size q is
served in at most 5 time units. Note that when all 4i are equal,
9i
every backlogged queue is served at the same rate and GPS
becomes a simple Processor Scheduling (PS).
When it is required to determine a session's route as well as
the weights on that route, it becomes the case of adaptive routing. We show that the adaptive routing case can be formulated
as a mixed integer linear program (MILP). The advantage of
both the LP and MILP is that their computational complexity is
independent of the number of sessions in the networks.
The second problem in this paper investigates the performance of PGPS scheduling policy when providing per-session
QoS guarantees. Here, session requests and terminations arrive
at random. We measure the performance in terms of weighted
carried traffic. We derive an upper bound on the weighted
carried traffic for any heuristic algorithm for admission control that operates within the locally stable domain. This upper
bound can be obtained by computing a simple linear program
(LP). By simulating a simple heuristic algorithm for admission control, we show that this upper bound is reasonably tight.
Thus our upper bound can be used as a metric against which
the performance of different algorithms can be compared.
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21 29
gE =
--+$
4l.
(2)
Cj=l4j
111. PGPS END-TO-ENDDELAYBOUND
Consider a set of sessions that are to be routed through a
network. All sessions are leaky bucket constrained with parameters (oi,
p i ) for session i . A session i is leaky bucket
constrained with parameters ( o i , p i ) if the amount of traffic that session i injects in the network, A i ( ~ , t during
),
any
interval(r,t], is bounded as A i ( ~ , t5) gi p i ( t - r),W 2
T >_ 0. The network is stable if the utilization of all links
is less than 1, where the utilization on link I, U' is defined as
+
ci':/')
U' =
1
'
, where 1(1)is the set of sessions that are going through link 1 ( [ 2 ] , [3]). Now consider a GPS assignment
for each session at all links on its path. We focus our attention
on a particular session i . This session traverses the network
through a set of I< links denoted as P ( i ) . Let the propagation delay along the path of session i be drrop. At each link
1 E P ( i ) ,the GPS server of that link guarantees a minimum
rate gl at which session i traffic is served. Note that with the
GPS assignment being known, the use of eqn (2) gives us g i .
Hence the minimum session i backlog clearing rate along its
route is gi = minl,p(i) gf To take packetization into account,
we consider the packetized version of GPS namely Packet GPS
(PGPS) in this paper. PGPS is a non-preemptive policy that
tracks GPS. Let Li be the size of maximum length packet from
session i and L be the size of maximum length packet among
all sessions. A session is defined to be locally stable if gi > pi.
Locally stable sessions have a simple and closed form bound
on end-to-end delay as stated in the following theorem [3].
Theorem 111.1: If gi 2 pi for session i , then
DT 5
oi
+ I<Li +
gi
{+dyP
/EP(i)
'
where 0,' is the maximum end-to-end delay that a session i
packet may suffer while traversing the network. In other words,
the GPS weight assignment provides an end-to-end delay guarantee of 05 for session i . Thus, given a predetermined endto-end delay bound Dreqd,we can use the above expression to
compute the minimum rate required to obey the delay bound
for session i . For example, for a voice session having c = 100
Bytes, p = 64 Kbps, P e q d = 50 ms, Li = 100 Bytes and
L = 1.5 KBytes, going over a 3-link route, gi turns out to be
85 Kbps. In this paper, the network operates under the locally
stable regime.
Class
1: voice
2: video conf
3: st video
a(kB)
0.1
IO
1 00
p(Mbps)
0.064
0.5
3
D(ms> L(kB)
50
0.1
75
1.5
100
1.5
IV. PGPS WEIGHTASSIGNMENT
PROBLEM
A. An LP Formulation f o r Fixed Routing
A session i is characterized by a 4-tuple ( a i , p i , D i ,Li)
where ai and p i are the leaky bucket parameters of session i ,
Dj is the maximum end-to-end delay requirement of session i
and Li is the maximum length of a packet of session i. Sessions
are grouped into traffic classes (e.g., table I). Sessions belonging to a traffic class have identical 4-tuples. Denote by C, M , L
the number of traffic classes, the number of available paths and
the number of links in the network respectively. A fixed routing
problem is one in which the path that a session is going to use
is prespecified. There is no choice as to which path should be
chosen for a session. Our aim is to check the schedulability of
a given set of sessions and to obtain a GPS assignment if the set
of sessions is schedulable. We now formulate this problem as
follows. bij is an indicator where bij = 1 if path i uses link j
and bij = 0 otherwise. nij is the number of class i sessions that
use path j . Without loss of generality, on a link, we assign the
same 4 to the sessions belonging to a class that use the same
path. Hence we can couple the q!J assignments for the sessions
of a class i on link l and having the same path into a single
4. Let & j l be this coupled assignment. 4 i j l is zero if class i
session doesn't use path j or if link 1 doesn't belong to path j .
Our linear formulation computes a q!Jij/ assignment. We make
use of the delay guarantee bound of theorem 111.1 to compute
the rate g i j needed for a class i session on path j to achieve
its delay bound. So the GPS assignment will be such that all
sessions are locally stable. Denote by r' the capacity of link 1.
The link capacities r1 and the bij and nij are given. The 4 i j l is
to be computed in such a way that all sessions get routed while
the utilization on each link remains less than 1 and the end-toend delay requirements are obeyed. So we have the following
linear formulation.
Find @ = ( & l ) such that:
4ijir'
2
nijbjl m a ( p i , g i j )
Vi E C,j E M,l E L
(3)
(4)
4ijl
5
nijbjl
vi E c,j E M,1 E L
(5)
(3) allocates so that the end-to-end delay requirements are
obeyed. (4) normalizes the q5 assignments at all links and with
#J
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02001 IEEE
Fig. 2. Example to illustrate q5 assignment in fixed routing case. Links are 155
Mbps and link delays are 4 ms. Sessions belong to the classes of table 1.
(a,,b ) and (c, d ) are the only SD-pairs on which sessions can be present
and there i s only 1 route per SD-pair which is prespecified. We allow 40
voice sessions, 2 vc sessions and 8 stv sessions on SD-pair ( a , b ) ; and 41
voice sessions, 2 vc sessions and 8 stv sessions on SD-pair ( e ,d ) . With
this, no more sessions can be admitted.
( 3 ) ,it keeps link utilization below 1. Note that there is no loss of
generality in this normalization. ( 5 )forces $ i j i to 0 at appropriate links. Any @ satisfying the above constraints gives us a GPS
assignment that achieves the end-to-end delay requirements of
the sessions. Note that this formulation is linear in 4 and a
assignment can be computed by solving a linear program (LP)
with above set of constraints. Also the computational complexity of the LP is independent of the number of sessions present
in the network.
We illustrate the usefulness of the L P formulation for a simple network of fig 2. All links have bandwidth = 155 Mbps and
link propagation delay = 4 ms. Assume ( a ,b ) and (e,d ) are the
only SD-pairs on which sessions can be present. The sessions
belong to the classes of table I. Session routes are fixed and
suppose sessions on ( a , b ) follow the route a -+ c + b and
sessions on (e,d ) is c -+ b -+ d. All routes span over two hops.
The rate required by a voice session is 64 Kbps as given by the
maximum of its p and rate computed from eqn 111.1. Similarly,
a video conferencing (vc) session requires a rate of 1.558Mbps
and a stored video (stv) session requires a rate of 8.9716 Mbps.
We consider the following set of sessions: On SD-pair ( a ,b ) ,
40 voice sessions, 2 vc sessions and 8 stv sessions; on SD-pair
(c, d), 41 voice sessions, 2 vc sessions and 8 stv sessions, that
is these are the nij for our example, see fig 2. By solving the
LP, it can be seen that the above 7 z i j set is a feasible set and the
respective 4 weights given by the LP are shown in table 11. We
also note that no more sessions can be admitted on any SD-pair;
if admitted, total session rate on any link exceeds the link rate.
In the next section, we show how adaptive routing can be used
to increase the schedulable region.
2130
TABLE I1
4 A S S I G N M E N T FOR
THE FIXED KOUTING h X A M P L E .
TABLE111
f#I A S S I G N M E N T F O R THE
I session
ADAPTIVI;. R O U T I N G
I
video conf
st video
EXAMPLE.
0.1 105
0.8689
0.0402
0.9264
0.1 104
0.8685
Fig. 3. Fixed routing example continued to illustrate routing & f#I assignment
in adaptive routing case. Here 2 routes per SD-pair are allowed. Because
of this, we can 10 voice sessions, 9 vc sessions and 7 stv sessions on the
new route for SD-pair ( a , b ) and 10 voice sessions, 9 vc sessions and 7 stv
sessions on the new route for SD-pair (c, d ) . This makes total 50 voice
sessions. 11 vc sessions and 15 stv sessions on SD-pair (a. b ) and total 51
voice sessions, 11vc sessions and 15 stv sessions on SD-pair ( c , d ) .
I
d on link
0.1806
0.8 1 1 1
B. Ail MILP Formulation for Adaptive Routing
In the adaptive routing case, it is not known which session
uses which path. But the source and destination of a session
are known. Also the paths that are available for a SD-pair are
known. The adaptive routing problem is to select paths for
the sessions such that the end-to-end delay requirements are
obeyed. Hence the adaptive routing formulation is based on
a class-SD-pair formulation. We define sij and a,ij as follows.
s i j is the number of class i sessions whose SD-pair is j . aij = 1
if path i is for SD-pair j . g i j , b i j , nij, @, r' are defined as in section IV-A. Note that, here @ as well as N are to be computed
in such a way that all sessions get routed while the utilization
on each link remains less than 1 and the end-to-end delay requirements are obeyed. N gives the routing information and
@ gives the GPS assignment. So we have the following linear
formulation.
Find N and
such that:
i=l j = 1
4ijl
5
nijbji
Vi E C , j
E M,l E L
(9)
Note that in the above formulation 4 are real nonnegative variables and n i j are integer variables. (6) is the only additional
constraint which decides the routing of sessions. The above
formulation is linear in 4 and nij . Since nij is an integer variable, @ and N assignments can be computed by a mixed integer
linear program (MILP) with above set of constraints. Note that
the computational complexity of the MILP is independent of
the number of sessions present in the network.
We continue with the example of last section to illustrate the
advantage of adaptive routing. We allow one more route per
SD-pair as follows. Sessions on ( a ,6 ) can also use a -+ d -+ 6
0-7803-7097-11011$10.00 02001 IEEE
2131
and sessions on ( c , d ) can also use c -+ a - + d. We add 10
voice sessions, 9 vc sessions and 7 stv sessions on the new route
for SD-pair ( a ,6 ) . Similarly we add 10 voice sessions, 9 vc
sessions and 7 stv sessions on the new route for SD-pair (c,d )
as in fig 3. So on SD-pair ( a , 6 ) total voice sessions are 50,
total vc sessions are 11 and total stv sessions are 15. On SDpair ( c ,d ) total voice sessions are 51, total vc sessions are 11
and total stv sessions are 15. This gives us the s i j and nij sets.
By solving the MILP, it can be seen that this is a feasible set.
The MILP also gives the 4 weights which are shown in table 111.
Also, we can add no more sessions of any class on any SD-pair.
Hence this set is an extreme point. Thus with adaptive routing
we were able to admit more sessions.
Note that in both routing cases, max(pi, g i j ) rate is allocated
to an admitted session. When gij > p i , additional bandwidth
= g i j - p i is allocated for a class i session. This additional
bandwidth remains unused and because of such sessions, network utilization may remain low. We call such sessions delay
constrained sessions. Thus a voice session of table I becomes
delay constrained when it has to go overmore than 2 hops. Both
video sessions of table I are delay constrained for any number
of hops. Let us consider 3 hop case. The rate required for
voice is 0.085 Mbps, for video conferencing 1.85 Mbps and for
stored video it is 9.53 Mbps. So 0.021 Mbps per voice session,
1.35 Mbps per video conf. session and 6.53 Mbps per stored
video session bandwidth remains unused. We note that this is
the price one has to pay for achieving delay guarantees under
PGPS scheduling.
V.
PERFORMANCE O F
PGPS
SCHEDULING
A. The Upper Bound
In this section we investigate the performance of PGPS policy under the locally stable regime. We assume the sessions
arrive and depart at random. The goal is to maximize the
weighted carried traffic, for a given offered load (in Erlangs)
of respective classes on each SD-pair. The weight reflects in
some sense the amount of resources (bandwidth) required for
that particular traffic class. The maximization problem is formulated as an Integer Linear Program (ILP). However, solving
an ILP is computationally expensive. So we relax the integer
constraints of the ILP to get a simple LP. The L P solution is an
upper bound on the ILP solution. We then go on to show that
this is a tight upper bound for any heuristic algorithm operating
in the locally stable domain.
Note that in the case of PGPS routing & weight assignment
problem, we were able to formulate fixed routing case as an LP
even though it was a special case of adaptive routing case. In
this section we only consider the adaptive routing case because
there is no gain in formulating the two cases separately. Fixed
routing case is treated as a special case of adaptive routing.
Before proceeding further we define a few terms here.
C ,M , P, L , ( a i j ) ,( b i j ) are defined as before. p i , oi, Di, Li are
the traffic descriptors and g i j is the rate required to obey endto-end delay bound for class i session on path j as before. Let
X denote the total offered load in Erlangs. The ( p i j ) matrix
gives the distribution of offered load among different classes
for the SD-pairs. So pijX is the offered load of class 1: traffic
on SD-pair j . The offered load for the static case (a fixed set
of sessions) is the number of sessions that are available to be
routed. In the dynamic case, it is the expected number number
of sessions that would be in progress if one could successfully
route all session arrivals. Let s i j denote the number of class i
sessions carried between SD-pair j . Also nij denotes the number of class i sessions carried on path j . In the dynamic case,
sij and n i j are the expectations of the respective quantities.
We measure the performance of PGPS policy under locally
stable regime in terms of weighted carried traffic. The weight
reflects in some manner the amount of resources (bandwidth)
required for that particular traffic class. We fix weight for traffic class i equal to p i . We want to find the optimal algorithm
for rate allocation so that the end-to-end delay bounds of admitted sessions are obeyed and the weighted carried traffic is
maximized. In the practically important dynamic case, where
sessions arrive at random, hold bandwidth along their paths for
random durations and then depart, we are interested in maximizing the weighted expected carried traffic. Then, the optimal
algorithm for the bandwidth allocation problem is found by by
solving the following ILP (integer linear program) whose value
we denote by I ( X , p ) .
(Maximize weighted carried traffic)
C
I ( X , P )=
P
m a x C ~ i C s i j
i=l
j=1
subject to
M
sij
=
nikakj
~i E C , Y E
~ P (10)
k=l
Sij
C
C
5
pijX
<
T'
'di€C,'dj E P
(1 1)
M
Cgijl~ijbj,
i=l j=1
Yl E L
(12)
where s i j , nij 2 0 and are integer variables.
Note that any heuristic algorithm for allocating rates under
the locally stable PGPS regime has to satisfy the constraints of
the above ILF? So the solution of above ILP produces an upper
bound on the weighted carried traffic of any heuristic algorithm
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-~
4
Fig. 4. Network to illustrate the performance of PGPS rate allocation. It has 5
nodes and 8 links. Links are 155 Mbps and link delays are 4 ms.
that operates within the locally stable domain. But solving ILP
may be computationally expensive and hence we are interested
in an easily computable upper bound. If we relax the integer
constraints of above ILP, we get an upper bound on the solution of ILP and hence an upper bound on the performance of
any heuristic algorithm for rate allocation that operates within
the locally stable domain. Let us denote this upper bound by
L(X,p ) . Though computationally inexpensive, the L P bound is
really useful if it is reasonably tight. In the next section, we
consider a heuristic algorithm for bandwidth allocation and we
show by simulations that the LP upper bound is indeed reasonably tight. This LP upper bound can be used as a metric
against which the performance of different algorithms can be
compared.
B. Comparison with a Heuristic Rate Allocation Algorithm
Here we show the utility of the LP upper bound that is derived in the last section. The network is shown in fig 4 and the
traffic descriptors are as in table 1. We consider the following
shortest-path heuristic bandwidth allocation algorithm. The set
of shortest paths between an SD-pair is ordered in some manner. The required rate to provide end-to-end delay guarantee for
the traffic of an arriving session is computed for the first path
on the list. If there is enough rate available along the path, the
session is admitted on that path. Otherwise, we check for the
second path of the list and so on. If no path can be found, the
session is considered blocked. We simulated the performance
of this simple algorithm for traffic where session requests are
assumed to arrive according to a Poisson process and last for a
duration that is exponentially distributed. We consider only the
case of uniform traffic, i.e., pij = 1/(CP) here although the
upper bound is applicable to any traffic pattern.
Fig 5-6 show the weighted carried traffic as a function of
offered traffic for above heuristic algorithm and their upper
bound. For example, in Fig 5 we compare the performance
of above heuristic where at most 2 shortest paths are allowed
between any SD-pair. It can be seen that the performance of
the heuristic is close to that achievable by anj~algorithm for
this network up to reasonably high offered load. It is possible
to conclude thus because our upper bound is reasonably tight
here. Moreover, we have observed that the offered load where
the upper bound is tight has reasonably low blocking probability. We would indeed like to operate in low blocking probability region. The part where the upper bound is not so tight is not
of much interest and hence we can safely conclude that over
region of interest, our upper bound is reasonably tight. Now
consider fig 7 which shows the performance bound of various
2132
c
optimal algorithms. It can be seen that we don’t gain much
by considering complicated algorithms which try to use many
paths between an SD-pair. In our example, the performance,
when at most 2 paths between any SD-pair are allowed, is almost the same as that obtained by considering all paths between
an SD-pair.
4
++
++++++
+++++++++++++++++
i
VI. C O N C L U S ~ O N
4
heuristic: 2
bound: 2
2
0
4
6
8
10
14
12
16
18
20
Offered Traffic (Erlangs)
Fig. 5. Weighted carried traffic of heuristic and its bound computed by the LP
are plotted. At most 2 paths between any SD-pair are considered. Heuristic
performs well for low offered traffic.
U
E
e
2.5 -
/Ir
. . . . . . . . . . . . . . . . . . . . . . .
:/++++
2 -
:+
1.5
-
j
I -:
i
0.5
0
heuristic: 3 paths
bound: 3 paths
I
I
I
I
I
2
4
6
8
IO
+
----~---
I
12
I
I
I
14
16
18
20
Offered Traffic (Erlangs)
Rg. 6. Weighted carried traffic of heuristic and its bound computed by the LP
are plotted. At most 3 paths between any SD-pair are considered. Heuristic
performs well for low offered traffic.
We have addressed the problem of routing sessions with delay and rate requirements in a network under PGPS scheduling.
In the first problem that we have considered, we try to find a
GPS weight assignment (4 assignment) for a given set of sessions so that their delay requirements are obeyed. Fixed routing
case and adaptive routing case are dealt separately. It is shown
that the fixed routing 4 assignment can be obtained by computing an LP. In the case of adaptive routing, it is shown that
the routing information and 4 assignment can be obtained by
computing an MILP. The computational complexity of both the
LP and MILP is independent of the number of sessions present
in the network. It is observed that for delay constrained sessions, PGPS allocates more additional bandwidth to achieve
delay guarantees. The links remain underutilized because of
this unused extra bandwidth.
The second problem that is addressed here is the performance of PGPS scheduling policy. We obtain an upper bound
on the performance of any heuristic bandwidth allocation algorithm that operates within the locally stable domain. The
bound can be obtained by solving an LP. The utility of our upper bound is shown by comparing it with the performance of
a simple shortest path heuristic algorithm. The performance of
this simple shortest path heuristic is close to that achievable by
aiiy algorithm. We can conclude thus because our upper bound
is reasonably tight here. Hence our upper bound can be used
as a metric against which the performance of different algorithms can be compared. Moreover, it can be seen from the
upper bound curves that the gain in weighted carried traffic is
not much for more number of paths between an SD-pair.
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bound: 2 paths
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[33
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--e---
a-----
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0-7803-7097-1/01/$10.00 02001 IEEE
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